Definition

The definition of (∞,1)(\infty,1)-sites parallels that of 1-categorical sites closely. In fact the structure of an (∞,1)(\infty,1)-site on an (∞,1)(\infty,1)-category is equivalent to that of a 1-categorical site on its homotopy category (see below).

For SS a sieve on cc and f:d→cf : d \to c a morphism into cc, we take the pullback sievef*Sf^* S on dd to be that spanned by all those morphisms into dd that become equivalent to a morphism in SS after postcomposition with ff.

A Grothendieck topology on the (∞,1)(\infty,1)-category CC is the specification of a collection of sieves on each object of CC – called the covering sieves , subject to the following conditions:

the trivial sieve covers – For each object c∈Cc \in C the overcategory C/cC_{/c} regarded as a maximal subcategory of itself is a covering sieve on cc. Equivalently: the monomorphism Id:j(c)→j(c)Id : j(c) \to j(c) covers.

the pullback of a sieve covers – If SS is a covering sieve on cc and f:d→cf : d \to c a morphism, then the pullback sieve f*Sf^* S is a covering sieve on dd. Equivalently, the pullback

a sieve covers if its pullbacks cover – For SS a covering sieve on cc and TT any sieve on cc, if the pullback sieve f*Tf^* T for every f∈Sf \in S is covering, then TT itself is covering.

An (∞,1)(\infty,1)-category equipped with a Grothendieck topology is an (∞,1)(\infty,1)-site.

Properties

Of sieves

Lemma

A sieve S′S' on cc that contains a covering sieve S⊂S′S \subset S' is itself covering.

Proof

For every f:d→cf : d \to c an object of S⊂C/cS \subset C_{/c}, the pullback sieve f*S′f^* S' equals the pullback sieve f*Sf^* S. So it covers dd by the second axiom on sieves. So by the third axiom S′S' itself is covering.

Proposition

There is a natural bijection between sieves on cc in CC and equivalence class of monomorphismsU→j(C)U \to j(C) in PSh(C)PSh(C).

Proof

First observe that equivalence classes of (−1)(-1)-truncated object of PSh(C/c)PSh(C_{/c}) are in bijection with sieves on cc:

An (∞,1)(\infty,1)-presheaf FF is (−1)(-1)-truncated if its value on any object is either the empty ∞-groupoid∅\emptyset or a contractible∞\infty-groupoid. The full subcategory of C/cC_{/c} on those objects on which FF takes a contractible value is evidently a sieve (because there is no morphism from a contractible to the empty ∞\infty-groupoid). Conversely, given a sieve SS on cc we obtain a (-1)-truncated presheaf fixed by the demand that it takes the value *=Δ[0]∈∞Grpd* = \Delta[0] \in \infty Grpd on those objects that are in SS, and ∅\emptyset otherwise.

Under this equivalence our bijection above maps to the statement that there is a bijection between sieves on cc and equivalence class of (−1)(-1)-truncated objects in PSh(C)/j(c)PSh(C)_{/j (c)}. But such a (-1)-truncated object is precisely a monomorphismU→j(c)U \to j(c).

Of coverages

Observation

The set of Grothendieck topologies on an (∞,1)(\infty,1)-category CC is in natural bijection with the set of Grothendieck topologies on its homotopy category.

Proof

Because picking full sub-1-categories as well as full sub-(∞,1)(\infty,1)-categories amounts to picking sub-sets/sub-classes of the set of equivalence classes of objects.

Corollary

If the (∞,1)(\infty,1)-category CC happens to be an ordinary category (for instance in its incarnation as a quasi-category it is the nerve of an ordinary category), then the structure of an (∞,1)(\infty,1)-site on it is the same as the 1-categorical structure of a site on it.

Examples

The trivial Grothendieck-topology on an (∞,1)(\infty,1)-category is that where the only covering sieve on each object cc is C/cC_{/c} itself. Equivalently, where the only covering monomorphisms U→j(c)U \to j(c) in PSh(C)PSh(C) are the equivalences.